Viscosity solutions to the degenerate oblique derivative problem for fully nonlinear elliptic equationsSolutions visqueuses du problème avec une dérivée oblique dégénérée pour une classe d'équations complètement non linéaires

In this paper we prove a comparison principle between the semicontinuous viscosity sub- and supersolutions of the tangential oblique derivative problem and the mixed Dirichlet–Neumann problem for fully nonlinear elliptic equations. By means of the comparison principle, the existence of a unique viscosity solution is obtained. To cite this article: P. Popivanov, N. Kutev, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 661–666.     

Nonhypoellipticity and comparison principle for partial differential equations of Black–Scholes type

This paper studies some less known properties of the Black–Scholes equation and of its nonlinear modifications arising in Finance. In particular, the nonhypoellipticity of the linear Black–Scholes equation is shown; a comparison principle is formulated for a class of nonlinear degenerate parabolic equations which incorporates the most relevant financial applications; finally, some comments on the properties of the viscosity solutions are given.     

Local solvability of semilinear partial differential equations

Summary It is proved an abstract theorem for local solvability of semilinear equations with complex coefficients. This result is illustrated by nonlinear perturbations of complex principal type and multiple complex characteristics operators.

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Riassunto Si dimostra un teorema astratto per la risolvibilità locale di equazioni semilineari di coefficienti complessi. Il risultato ottenuto viene illustrato mediante perturbationi nonlineari di operatori di tipo principale complesso e caratteristiche compless multiple.

     

Protter-Morawetz multidimensional problems

About 50 years ago M.H. Protter introduced boundary value problems that are multidimensional analogues of the classical plane Morawetz problems for equations of mixed hyperbolic-elliptic type that model transonic fluid flows. Up to now there are no general existence results for the Protter-Morawetz multidimensional problems, and an understanding of the situation is not at hand. At the same time, Protter also formulated boundary value problems in the hyperbolic part of the domain??the nonhomogeneous wave equation is studied in a (3+1)-D domain bounded by two characteristic cones and a non-characteristic ball. These problems could be considered as multidimensional variants of the Darboux problem in ?². In the frame of classical solvability the hyperbolic Protter problem is not Fredholm, because it has an infinite-dimensional cokernel. On the other hand, it is known that the unique generalized solution of a Protter problem may have a strong power-type singularity even for some very smooth right-hand side functions. This singularity is isolated at the vertex O of the boundary light cone and does not propagate along the characteristic cone. In the general case of smooth right-hand side function, some necessary and sufficient conditions for the existence of a bounded solution are given and a priori estimates for the solution are found. The semi-Fredholm solvability of the problem is proved.     

Gevrey hypoellipticity and solvability on the multidimensional torus of some classes of linear partial differential operators

Abstract In this paper we consider the problem of global analytic and Gevrey hypoellipticity and solvability for a class of partial differential operators on a torus. We prove that global analytic and Gevrey hypoellipticity and solvability on the torus is equivalent to certain Diophantine approximation properties. Keywords: Global hypoellipticity, Global solvability, Gevrey classes, Diophantine approximation property Mathematics Subject Classification (2000): 35D05, 46E10, 46F05, 58J99     

Compact travelling waves and peakon type solutions of several equations of mathematical physics and their Cellular Neural Network realization

This paper deals with compact travelling waves and peakon type solutions of several equations of mathematical physics and their Cellular Neural Network (CNN) realization. More precisely, we study different generalizations of the Camassa–Holm equation, of the Korteweg–de Vries equation and the nonlinear PDE describing the vibrations of a chain of particles interconnected by springs. In many cases the waves develop cusp type singularities at the peaks. In the second part of the paper the CNN realization of the compact travelling waves is fulfilled and the corresponding geometrical illustrations of the interaction of these waves are given.     

Existence results for second order boundary value problems using the barrier strip technique

Using barrier strip type arguments we investigate the existence of solutions of the boundary value problem ${x''=f(t,x),\;t\in(0,1),\;x(0)=A,\;x'(1)=0,}Using barrier strip type arguments we investigate the existence of solutions of the boundary value problem x"=f(t,x),  t ? (0,1),  x(0)=A,  x￠(1)=0,{x'=f(t,x),\;t\in(0,1),\;x(0)=A,\;x'(1)=0,} where the scalar function f(t, x) may be singular at x = A.     

Critical Gevrey index for hypoellipticity of parabolic operators and Newton polygones

Summary In this paper we give geometrical expressions of the (non) hypoellipticity in Gevrey spaces of parabolic operators via Newton polygones. We also determine the critical Gevrey class for which the hypoellipticity holds.Partially supported by GNAFA, CNR, Italy.Partially supported by JSPS, Japan and a grant MM-410/94 with MES, Bulgaria.Partially supported by Chuo University special research fund.